Theorem fmf 23954

Description: Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fmf	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Proof of Theorem fmf

Dummy variables 𝑓 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7465	. . . 4 ⊢ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))) ∈ V
2		eqid 2736	. . . 4 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
3	1, 2	fnmpti 6710	. . 3 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) Fn (fBas‘𝑌)
4		df-fm 23947	. . . . . 6 ⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))))
5	4	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))))))
6		dmeq 5913	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
8		fdm 6744	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
9	8	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
10	7, 9	sylan9eqr 2798	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝑌)
11	10	fveq2d 6909	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (fBas‘dom 𝑓) = (fBas‘𝑌))
12		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
13		imaeq1 6072	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓 “ 𝑦) = (𝐹 “ 𝑦))
14	13	mpteq2dv 5243	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
15	14	rneqd 5948	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
16	12, 15	oveqan12d 7451	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
17	16	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
18	11, 17	mpteq12dv 5232	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
19		elex 3500	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
20	19	3ad2ant1 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ V)
21		fex2 7959	. . . . . 6 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝐹 ∈ V)
22	21	3com13 1124	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ V)
23		fvex 6918	. . . . . . 7 ⊢ (fBas‘𝑌) ∈ V
24	23	mptex 7244	. . . . . 6 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V
25	24	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V)
26	5, 18, 20, 22, 25	ovmpod 7586	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
27	26	fneq1d 6660	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ↔ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) Fn (fBas‘𝑌)))
28	3, 27	mpbiri 258	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
29		simpl1 1191	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋 ∈ 𝐴)
30		simpr 484	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
31		simpl3 1193	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
32		fmfil 23953	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
33	29, 30, 31, 32	syl3anc 1372	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
34	33	ralrimiva 3145	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
35		ffnfv 7138	. 2 ⊢ ((𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋) ↔ ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋)))
36	28, 34, 35	sylanbrc 583	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ↦ cmpt 5224  dom cdm 5684  ran crn 5685   “ cima 5687   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  fBascfbas 21353  filGencfg 21354  Filcfil 23854   FilMap cfm 23942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-fg 21363  df-fil 23855  df-fm 23947

This theorem is referenced by:  rnelfm  23962